{
  "abstract": "We construct a family of Lax representations for the three-dimensional incompressible Navier--Stokes equations in vorticity form. The key idea is to introduce an auxiliary vector field, termed the \\emph{vorticity-diffusion potential}, which recasts the viscous Laplacian diffusion term as a Lie-bracket advection term. On solutions of the defining equation for this potential, the viscous vorticity equation takes the form of a pure Lie-transport equation along a modified divergence-free velocity field, while the defining constraint itself is encoded as the curl-free compatibility condition for an auxiliary scalar potential. We further show that the reduction of this three-dimensional system to two spatial dimensions yields a Lax representation for the two-dimensional Navier--Stokes vorticity equation, which we generalize to a family parametrized by an arbitrary smooth function of one variable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26668",
  "author_total": 1,
  "authors": [
    {
      "name": "Oleg I. Morozov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6aw84mfg7tkm1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26668",
  "fulltext_url": null,
  "id": "6aw84mfg7tkm1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26668",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7223021467",
  "pdf_url": "https://arxiv.org/pdf/2610.26668",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6aw84mfg7tkm1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2610.26668",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26668"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Lax representations for the Navier--Stokes equations",
  "title_text": "Lax representations for the Navier--Stokes equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:41Z",
  "url": "https://research.draconitic.com/paper/6aw84mfg7tkm1",
  "venue": "arXiv",
  "versions": []
}